@Iszi “The attacker now has an easier job: he can find any password whose hash begins with 2c26b46b68ffc68ff99b453c1d304134.”
and “The only way to invert a proper hash function is by straight brute force: guess the password, compute your guess's hash, compare with the reference hash. If the reference hash is a truncation of the real hash, the attacker gains a negligible amount of time doing the comparison. For a massive brute force attack, or if you truncate the hash too much, the attacker may have its job made easier by finding a collision on the truncated hash which wouldn't have been one on the original hash.”
@Gilles Yeah, that does a good job of explaining how the attacker's job is easier. (Though, for me, there are some readability issues.) However, what I was interested to see, and @ThomasPornin did a good job of spelling out in his edit, was a direct answer to the original headline: "Does truncating the hash make the password harder or impossible to crack?" Targeting the actual password, the answer is yes.
Addressing the latter part of the question though "are there problems with this" is what most of the answers have done - and the answer to that is absolutely, because truncating the hash means that there are a whole lot of other passwords that will now be valid.
So: the attacker will look for the password by making a guess, computing truncate(hash(guess)), and try again if truncate(hash(guess)) != reference_truncated_hash
the attacker stops when truncate(hash(guess)) == reference_truncated_hash
if hash is a decent cryptographic hash function (e.g. one of the SHA), the password is memorable by a human, and the length of the truncation is decent (e.g. 16 bytes), then I claim that it is
In an ideal hash, all the bits are independent: if you have 3 bits of the hash, and you're looking for a preimage, the best you can do is discard 1 guess in 8
